On the total curvatures of a tame function

Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities

World modeling in robotics

Abstract only

Tame Functions with strongly isolated singularities at infinity: a tame version of a Parusinski's Theorem

Let f be a definable function, enough differentiable. Under the condition of having strongly isolated singularities at infinity at a regular value c we give a sufficient condition expressed in terms of the total absolute curvature function to ensure the local triviality of the function f over a neighbourhood of c and doing so providing the tame version of Parusinski's Theorem on complex polynomials with isolated singularities at infinity.Comment: 20 page

Towards a cloud‑based automated surveillance system using wireless technologies

Cloud Computing can bring multiple benefits for Smart Cities. It permits the easy creation of centralized knowledge bases, thus straightforwardly enabling that multiple embedded systems (such as sensor or control devices) can have a collaborative, shared intelligence. In addition to this, thanks to its vast computing power, complex tasks can be done over low-spec devices just by offloading computation to the cloud, with the additional advantage of saving energy. In this work, cloud’s capabilities are exploited to implement and test a cloud-based surveillance system. Using a shared, 3D symbolic world model, different devices have a complete knowledge of all the elements, people and intruders in a certain open area or inside a building. The implementation of a volumetric, 3D, object-oriented, cloud-based world model (including semantic information) is novel as far as we know. Very simple devices (orange Pi) can send RGBD streams (using kinect cameras) to the cloud, where all the processing is distributed and done thanks to its inherent scalability. A proof-of-concept experiment is done in this paper in a testing lab with multiple cameras connected to the cloud with 802.11ac wireless technology. Our results show that this kind of surveillance system is possible currently, and that trends indicate that it can be improved at a short term to produce high performance vigilance system using low-speed devices. In addition, this proof-of-concept claims that many interesting opportunities and challenges arise, for example, when mobile watch robots and fixed cameras would act as a team for carrying out complex collaborative surveillance strategies.Ministerio de Economía y Competitividad TEC2016-77785-PJunta de Andalucía P12-TIC-130

Monotone functions and maps

In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in Basu et al. for semi-monotone sets.Comment: 30 pages. Version 2 appeared in RACSAM. In version 3 Corollaries 1 and 2 were corrected. In version 4 Theorem 3 is correcte

Euler-Bessel and Euler-Fourier Transforms

We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible \zed-valued functions to continuous $\real$-valued functions over a vector space. Core contributions include: the definition of the topological Bessel transform; a relationship in terms of the logarithmic blowup of the topological Fourier transform; and a novel Morse index formula for the transforms. We then apply the theory to problems of target reconstruction from enumerative sensor data, including localization and shape discrimination. This last application utilizes an extension of spatially variant apodization (SVA) to mitigate sidelobe phenomena

Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated Deduction, 2015. Proceedings to be published by Springer-Verla

\Lambda-buildings and base change functors

We prove an analog of the base change functor of \Lambda-trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine building is closed under ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings.Comment: revised version, 29 pages, to appear in Geom. Dedicat